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Exponential Functions and Models - PowerPoint PPT Presentation

Exponential Functions and Models. Lesson 5.3. Contrast. Definition. An exponential function Note the variable is in the exponent The base is a C is the coefficient, also considered the initial value (when x = 0). Explore Exponentials.

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Exponential Functions and Models

Lesson 5.3

• An exponential function

• Note the variable is in the exponent

• The base is a

• C is the coefficient, also considered the initial value (when x = 0)

• Given f(1) = 3, for each unit increase in x, the output is multiplied by 1.5

• Determine the exponential function

• Graph these exponentials

• What do you think the coefficient C and the base a do to the appearance of the graphs?

• Suppose you have a choice of two different jobs at graduation

• Start at \$30,000 with a 6% per year increase

• Start at \$40,000 with \$1200 per year raise

• Which should you choose?

• One is linear growth

• One is exponential growth

• How do we get each nextvalue for Option A?

• When is Option A better?

• When is Option B better?

• Rate of increase a constant \$1200

• Rate of increase changing

• Percent of increase is a constant

• Ratio of successive years is 1.06

• Consider a savings account with compounded yearly income

• You have \$100 in the account

• You receive 5% annual interest

View completed table

• Completed table

• Table of results from calculator

• Set Y= screen y1(x)=100*1.05^x

• Choose Table (♦Y)

• Graph of results

• Lesson 5.3A

• Page 415

• Exercises 1 – 57 EOO

• Consider an amount A0 of money deposited in an account

• Pays annual rate of interest r percent

• Compounded m times per year

• Stays in the account n years

• Then the resulting balance An

• Population growth often modeled by exponential function

• Half life of radioactive materials modeled by exponential function

• Recall formulanew balance = old balance + 0.05 * old balance

• Another way of writing the formulanew balance = 1.05 * old balance

• Why equivalent?

• Growth factor: 1 + interest rate as a fraction

• Consider a medication

• Patient takes 100 mg

• Once it is taken, body filters medication out over period of time

• Suppose it removes 15% of what is present in the blood stream every hour

Fill in the rest of the table

What is the growth factor?

• Completed chart

• Graph

Growth Factor = 0.85

Note: when growth factor < 1, exponential is a decreasing function

• For our medication example when does the amount of medication amount to less than 5 mg

• Graph the functionfor 0 < t < 25

• Use the graph todetermine when

• All exponential functions have the general format:

• Where

• A = initial value

• B = growth rate

• t = number of time periods

• When B > 1

• When B < 1

Using e As the Base

• We have used y = A * Bt

• Consider letting B = ek

• Then by substitution y = A * (ek)t

• Recall B = (1 + r) (the growth factor)

• It turns out that

• The constant k is called the continuous percent growth rate

• For Q = a bt

• k can be found by solving ek = b

• Then Q = a ek*t

• For positive a

• if k > 0 then Q is an increasing function

• if k < 0 then Q is a decreasing function

• For Q = a ek*t Assume a > 0

• k > 0

• k < 0

• For the functionwhat is thecontinuous growth rate?

• The growth rate is the coefficient of t

• Growth rate = 0.4 or 40%

• Graph the function (predict what it looks like)

• Change to the form Q = A*Bt

• We know B = ek

• Change to the form Q = A*ek*t

• We know k = ln B (Why?)

• May be a better mathematical model for some situations

• Bacteria growth

• Decrease of medicine in the bloodstream

• Population growth of a large group

• A population grows from its initial level of 22,000 people and grows at a continuous growth rate of 7.1% per year.

• What is the formula P(t), the population in year t?

• P(t) = 22000*e.071t

• By what percent does the population increase each year (What is the yearly growth rate)?

• Use b = ek

• Lesson 5.3B

• Page 417

• Exercises 65 – 85 odd